Explain Derangements. Write formula and give examples.

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Solution

A derangement is a permutation in which none of the objects appear in their "natural" (i.e., ordered) place. For example, the only derangements of { $1, 2, 3$ } are { $2, 3, 1$ } and { $3, 1, 2$ }. Similarly, the derangements of { $1, 2, 3, 4$ } are { $2, 1, 4, 3$ }, { $2, 3, 4, 1$ }, { $2, 4, 1, 3$ }, { $3, 1, 4, 2$ }, { $3, 4, 1, 2$ }, { $3, 4, 2, 1$ }, { $4, 1, 2, 3$ }, { $4, 3, 1, 2$ }, and { $4, 3, 2, 1$ }.

Derangements are permutations without fixed points (i.e., having no cycles of length one).

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